On a nonlinear stochastic integral equation of the Hammerstein type

Padgett, W. J.

doi:10.1090/S0002-9939-1973-0320663-2

On a nonlinear stochastic integral equation of the Hammerstein type
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by W. J. Padgett

Proc. Amer. Math. Soc. 38 (1973), 625-631

DOI: https://doi.org/10.1090/S0002-9939-1973-0320663-2

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Abstract:

A nonlinear stochastic integral equation of the Hammerstein type in the form \[ x(t;\omega ) = h(t;\omega ) + \int _s {k(t,s;\omega )f(s,x(s;\omega )} )d\mu (s)\] is studied where $t \in S,a$, a $\sigma$-finite measure space with certain properties, $\omega \in \Omega$, the supporting set of a probability measure space $(\Omega ,A,P)$, and the integral is a Bochner integral. A random solution of the equation is defined to be a second order vector-valued stochastic process $x(t;\omega )$ on $S$ which satisfies the equation almost certainly. Using certain spaces of functions, which are spaces of second order vector-valued stochastic processes on $S$, and fixed point theory, several theorems are proved which give conditions such that a unique random solution exists.

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